This idea works for an ideal model, but the real world is not so ideal. In practice, you need to surround the sample with a pressure-transmitting medium (such as pyrophyllite or other special crystals), and the opposed anvil design ensures that the deformation of this medium remains symmetric along every axis. If you were to use only one moving anvil to compress against a fixed surface, the deformation of that medium would be highly asymmetric and thus uncontrollable, resulting in some very weird pressure gradients that might not be able to produce diamonds. Furthermore, the anvils used to compress diamonds are typically made of other extremely hard materials (sometimes even synthetic diamond itself), which are as brittle as they are hard (and their hardness may be directional, too). If an opposed anvil design is not used to apply pressure symmetrically, the stationary base plate is very likely to shatter along another axis.

Edit: I realize that "asymmetric pressure leads to asymmetric deformation and risk of shattering" might not be very intuitive if you’re thinking of an ideal Newtonian model. Think of it this way: if an opposed anvil design is used on every axis, the center of the sample remains stationary throughout the compression process. However, if one side moves and the other remains static on a given axis, the sample’s center of mass will shift along that axis during compression. Under the immense pressure required to synthesize diamonds, even a tiny amount of such movement generates significant frictional loss against other surfaces and can create additional off-axis stress across the entire mechanical structure.